In the xy-plane, the y-coordinate of the y-intercept of the graph of the function f is c. Which of the...

1. TRANSLATE the problem information

• Given information:
  • The y-coordinate of the y-intercept of function \(\mathrm{f}\) is \(\mathrm{c}\)
  • Need to find which expression equals \(\mathrm{c}\)
• What this tells us: We need to connect the geometric concept of y-intercept to function notation

2. INFER what y-intercept means mathematically

• A y-intercept is where the graph crosses the y-axis
• The y-axis is the vertical line where \(\mathrm{x = 0}\)
• So the y-intercept occurs at the point \((0, \text{y-coordinate})\)

3. INFER the coordinate pair for this y-intercept

• Since the y-coordinate of the y-intercept is \(\mathrm{c}\)
• The y-intercept point must be \((0, \mathrm{c})\)

4. INFER the connection to function notation

• Function notation \(\mathrm{f(x)}\) means "the y-value when x is the input"
• Since our y-intercept is at \(\mathrm{x = 0}\) with y-coordinate \(\mathrm{c}\)
• This means \(\mathrm{f(0) = c}\)

Answer: A. \(\mathrm{f(0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse y-intercept with x-intercept, thinking the y-intercept occurs where \(\mathrm{y = 0}\) instead of where \(\mathrm{x = 0}\).

This fundamental misunderstanding leads them to think about where the function crosses the x-axis instead of the y-axis. They might randomly guess among the choices since their reasoning foundation is incorrect, potentially selecting any of Choice B, C, or D.

Second Most Common Error:

Missing conceptual knowledge: Students don't understand that function notation \(\mathrm{f(0)}\) represents the y-value when \(\mathrm{x = 0}\).

Even if they correctly identify that the y-intercept occurs at \(\mathrm{x = 0}\), they fail to make the connection that \(\mathrm{f(0)}\) gives exactly this y-value. This leads to confusion about what the answer choices represent, causing them to get stuck and guess randomly.

The Bottom Line:

This problem tests the crucial connection between geometric concepts (intercepts) and algebraic notation (function values). Success requires both understanding what y-intercept means geometrically AND knowing how function notation works.